Distributed Greedy Approximation to Maximum Weighted
Independent Set for Scheduling with Fading Channels
Changhee Joo
Xiaojun Lin
Jiho Ryu
ECE, UNIST
UNIST-gil 50
Ulsan, South Korea
ECE, Purdue University
465 Northwestern Ave.
West Lafayette, IN
47907-2035
ECE, UNIST
UNIST-gil 50
Ulsan, South Korea
[email protected]
[email protected]
Ness B. Shroff
[email protected]
ECE and CSE, OSU
2015 Neil Ave.
Columbus, OH 43210
[email protected]
ABSTRACT
Categories and Subject Descriptors
Developing scheduling mechanisms that can simultaneously
achieve throughput optimality and good delay performance
often require solving the Maximum Independent Weighted
Set (MWIS) problem. However, under most realistic network settings, the MWIS problem can be shown to be NPhard. In non-fading environments, low-complexity scheduling algorithms have been provided that converge either to
the MWIS solution in time or to a solution that achieves
at least a provable fraction of the achievable throughput.
However, in more practical systems the channel conditions
can vary at faster time-scales than convergence occurs in
these lower-complexity algorithms. Hence, these algorithms
cannot take advantage of the opportunistic gain, and may
no longer guarantee good performance. In this paper, we
propose a low-complexity scheduling scheme that performs
provably well under fading channels and is amenable to implement in a distributed manner. To the best of our knowledge, this is the first scheduling scheme under fading environments that requires only local information, has a low
complexity that grows logarithmically with the network size,
and achieves provable performance guarantees (which is arbitrarily close to that of the well-known centralized Greedy
Maximal Scheduler). Through simulations we verify that
both the throughput and the delay under our proposed distributed scheduling scheme are close to that of the optimal
solution to MWIS. Further, we implement a preliminary version of our algorithm in a testbed by modifying the existing
IEEE 802.11 DCF. The preliminary experiment results show
that our implementation successfully accounts for wireless
fading, and attains the opportunistic gains in practice, and
hence substantially outperforms IEEE 802.11 DCF.
C.2.1 [Network Architecture and Design]: Wireless communication; G.2.2 [Graph Theory]: Network problems
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Keywords
Distributed algorithm, wireless scheduling, maximum weighted
independent set
1. INTRODUCTION
Scheduling is one of the most fundamental functionalities
of wireless networks. It determines which links should transmit at what time and at what data rate. It is well-known
that solving the scheduling problem is inherently difficult because the interference relationship is often non-convex and
even combinatorial in nature. Further, for large networks it
is imperative that the scheduling algorithm is of low complexity and can be implemented in a fully distributed manner. Such requirements make it highly challenging to design
easy-to-implement scheduling algorithms.
In the literature, it is well-known that the so-called MaxWeight
algorithm is throughput optimal [27]. For graph based interference models, where whether two links interfere or not can
be specified by a binary parameter, the MaxWeight algorithm corresponds to the solution to a Maximum Weighted
Independent Set (MWIS) problem in the conflict (or interference) graph as follows. In the conflict graph, each link is
mapped onto a vertex and two vertices (links) that interfere
with each other are connected by an edge. A set of nonconnected vertices, which is called an independent set, can
transmit data simultaneously. Further, each vertex of the
conflict graph is given a weight, which is typically the product of the link rate and its queue length, and which varies
across time due to changing queue lengths and time-varying
channels. The MaxWeight algorithm then computes an independent set that has the largest total weight (i.e., solution
to the MWIS problem). Although the MaxWeight algorithm
is throughput optimal, the MWIS problem is NP-Hard in
general [13]. Hence, the MaxWeight algorithm incurs high
complexity, and further, it is a centralized algorithm that requires global information. Thus, the MaxWeight algorithm
is not amenable to practical implementation.
In the literature, there have been many efforts to develop
low-complexity and distributed scheduling algorithms with
provably good throughput performance [6,8,11,12,14,16,17,
20,21,23]. These algorithms differ in terms of their throughput guarantee, complexity, and delay performance. They
can be classified into two categories, depending on whether
or not they account for channel fading.
There have been many more scheduling solutions for wireless systems without fading. Low-complexity scheduling algorithms have been developed with complexity that grows
significantly slower than the network size, and can yet guarantee a non-negligible fraction of the optimal system capacity. As a point of comparison, the Greedy Maximal Scheduling (GMS) algorithm (also known as Longest Queue First
(LQF) algorithm) can provably attain a fraction of the optimal capacity, with complexity that grows linearly with the
total number of links L [5]. Other algorithms can reduce the
complexity even further. For example, the Maximal Schedul1
ing algorithm can attain at least ∆
of the optimal capacity,
with O(log N ) complexity [29], where ∆ denotes the maximum conflict degree (see (2) for the definition) and N denotes the number of nodes. The Constant-time scheduling
algorithms, instead, can achieve a comparable capacity with
O(1) complexity, i.e., the complexity does not grow with the
network size [17]. Further, both the Pick-and-Compare algorithm [6,21] and Carrier Sensing Multiple Access (CSMA)
algorithm [11, 23] have been shown to achieve the optimal
throughput. They incur O(L) and O(1) complexity, respectively. We note that these two algorithms have been observed to lead to poor delay performance [8, 20], and hence
the utility of the throughput gain may be debatable, especially for delay-sensitive applications. Nonetheless, these
results indicate that good throughput performance may be
attained for non-fading environments using algorithms with
very low complexity.
In practice, however, most wireless systems experience
some level of channel fading. When link rates vary across
time due to fading, the system throughput can be further improved by scheduling links when their rate are high. This is
known as the opportunistic gain [19]. Exploiting opportunistic gain has been extremely popular in cellular systems. For
ad hoc wireless networks, the MaxWeight algorithm can exploit this opportunistic gain and in fact achieve the optimal
throughput even with fading. However, many of the lowcomplexity scheduling algorithms described in the previous
paragraph cannot exploit the opportunistic gain, and their
performance in fading environments will be much worse.
Take CSMA and Pick-and-Compare algorithms as examples. They reduce complexity by amortizing the computation across many time-slots, and hence need to take many
iterations to find a close-to-optimal schedule. In fading environments, the link rates could have changed significantly
before these algorithms can find a good schedule. Hence,
they will not be able to exploit the opportunistic gain unless the fading is very slow [30]. Similarly, it appears to
be difficult for the Maximal Scheduling algorithm and the
Constant-time scheduling algorithm to account for channel fading and still guarantee a provable fraction of the
optimal capacity. Recently, there have been a few other
low-complexity schemes that are provably efficient with fading channels. However, they are either limited to singlehop networks [16] or their performance guarantees are much
lower [12].
An exception is perhaps the GMS scheduling algorithm,
which computes an approximation to the MWIS problem
by choosing the highest weight vertex first, and can guar1
antee ∆
fraction of the optimal capacity in both fading and
non-fading environment. Other greedy approximations have
also been proposed in [24, 28]. However, they require centralized operations and linear complexity O(L). Although
distributed greedy approximation algorithms have been developed [4, 10], they still incur a worst case time-complexity
of O(L). This high complexity has become a major obstacle preventing these algorithms from being used in practical system because the channel conditions can vary at faster
time-scales than O(L). Given that fading is a prevalent phenomenon in most modern wireless systems, an interesting
open question is how one can develop distributed scheduling
algorithms with even lower complexity and yet guarantee
good performance .
In this paper, we answer this open question by proposing a
novel low-complexity and distributed greedy approximation
algorithm, called DistGreedy, for both fading and non-fading
environments. In contrast to the known greedy approximations [4, 10, 24, 28], our proposed DistGreedy algorithm incurs a low logarithmic complexity O(log L) that grows slowly
with the network size. Further, it requires only local information (such as queue length and link rates of neighboring links), and can be implemented in a distributed fashion.
We analytically show that our low-complexity distributed
1
-approximation
algorithm produces a schedule that is a ∆
to the MWIS problem, and show through simulations that
DistGreedy often achieves scheduling performance far better than the provable bounds. Indeed, it empirically achieves
throughput and delay performance that is close to that of the
MaxWeight scheduler. We also conduct preliminary experiments with implementation in a real testbed. We implement
a new MAC protocol that captures the essence of DistGreedy
by modifying the IEEE 802.11 DCF. Performance comparison with the IEEE 802.11 DCF under channel fading shows
that the DistGreedy algorithm can exploit the opportunistic
gains and thus substantially outperform IEEE 802.11 DCF
in fading environments.
The rest of the paper is organized as follows. The system
model is described in Section 2. The DistGreedy algorithm
is proposed and analyzed in Section 3. We numerically evaluate its performance in Section 4, and provide preliminary
experiment results based on a testbed implementation in
Section 5. Then, we conclude.
2. SYSTEM MODEL
We consider a wireless network with N nodes and L directed links. We assume that time is slotted and that a
single frequency channel is shared by all the links. Multiple
link transmissions at the same time slot may fail due to wireless interference. We assume that there is no link error, i.e.,
a link transmission is successful if there is no simultaneous
interfering transmission.
The link rate of a successful transmission depends on its
channel state. We assume that the channel state is fixed
during a time slot, and changes across time slots. We denote
the (global) channel state by h. when the channel is in state
h, link l can transfer rlh unit of data if its transmission is
successful. Let H denote the set of all the channel states.
We assume that the channel state
Phas a finite space with a
stationary distribution π h , with h∈H π h = 1.
In order to account for wireless fading, we employ a channelh
dependent interference model as follows. Let Ckl
∈ {0, 1}
denote the interference relationship between link k and link
h
l when the channel state is h. We set Ckl
= 0 if link l does
not interfere with link k (and therefore they can transmit
h
simultaneously), and Ckl
= 1, otherwise. We assume that
h
h
the interference relationship is symmetric, i.e., Ckl
= Clk
.
We note that the dependency on h represents a major departure from existing works for non-fading environments.
Specifically, the interference relationship as well as the link
rate may change across time in our model. This model is
not only a simplified version that captures the fundamental characteristics of the more accurate SINR interference
model [9], but also a general model that includes many interference models used in the literature to model FH-CDMA,
Bluetooth, and IEEE 802.11 DCF network systems [25, 29],
as special cases.
Given a network system, our interference model admits a
unique conflict graph at each channel state h, which clearly
presents the underlying interference constraints. For each
link l ∈ L, we draw a vertex in the conflict graph, which is
also denoted by the same alphabet l. For every two vertices
h
k, l with Ckl
= 1, we connect them with an edge in the
conflict graph. Let Gh = (V, E h ) denote the conflict graph
with the set V of vertices and the set E h of edges under
channel state h. The conflict graph explicitly shows the
interference relationship of any two vertices (i.e., links in the
original network). In the sequel, we deal with the conflict
graph throughout the paper.
We now formally formulate the Maximum Weighted Independent Set (MWIS) problem. Suppose that the channel
state is h at time slot t. We consider the conflict graph Gh
constructed from the interference constraints under channel
state h. We begin with some definitions. Vertex x is a neighbor of vertex v, if they are connected by an edge in the conflict graph. Let I h (v) denote the set of neighbors of vertex v
including v, and let I h (A) denote the set of neighbors of vertices in A, i.e., I h (A) := ∪v∈A I h (v). Let wv (t, h) denote a
weight associated with vertex v. In particular, we define the
weight of vertex v as the product of queue length Qv (t) and
transmission rate rvh . Let w∗ (t, h) denote the largest weight,
i.e., w∗ (t, h) := maxv∈V wv (t, h). Further, let w̄v (X; t, h)
denote the largest weight in the neighborhood of vertex v
within X, i.e., w̄v (X; t, h) := maxx∈X∩I h (v) wx (t, h).
We say that a set S of vertices is an independent set (or
a feasible schedule) if no two vertices in the set are neighbors. Further, an independent set is maximal if no extra
vertex can be added. Such an independent set is also called
a maximal matching. Let Sh denote the collection of all
the feasible independent sets that are available in Gh . The
MWIS problem can be formulated as finding S ∗ such that
X
S ∗ ∈ argmax
wv (t, h).
(1)
S∈Sh
v∈S
It has been known that at each time t, given a channel state
h, the solution to the MWIS problem with weight wv (t, h) =
Qv (t)·rvh results in a throughput-optimal MaxWeight scheduling scheme [22]. However, due to the high computational
complexity and the requirement of global information, such
a MaxWeight algorithm is difficult to implement in practice. On the other hand, it has been shown in [15, 18] that
an imperfect scheduling solution that solves (1) within a factor of γ at every time t achieves at least γ fraction of the
optimal throughput. To this end, our goal is to develop practical low-complexity scheduling algorithm that can approximately solve (1) with a provable fraction in a distributed
fashion.
Remarks: In the above MaxWeight scheduling scheme, we
implicitly assume single-hop traffic, i.e., packets are transmitted over a single link and leave the system immediately
after the transmission. For multi-hop traffic, the same MaxWeight
algorithm can be used by replacing the queue length with a
queue differential. (See [27] for the details.) Similarly, our
DistGreedy algorithm described in the next section can be
extended to multi-hop scenarios in a straightforward manner.
Finally, we define the vertex degree δ(h) := maxv∈V |I h (v)|,
where | · | denotes the cardinality of the set, and the maximum conflict (or interference) degree ∆(h) as
∆(h) := max |I h (v) ∩ S|.
v∈V,S∈S
(2)
In the network, the maximum conflict degree represents the
maximum number of simultaneous transmissions in the neighborhood of any link, which can be upper bounded by a constant in many practical interference models [7, 9]. Also, we
define ∆∗ = maxh∈H ∆(h).
3. DISTRIBUTED GREEDY APPROXIMATION
In this section, we describe our distributed approximate
solution to (1) and analyze its performance. We emphasize
that the algorithm operates in a distributed manner and
each vertex (link) requires only local information from its
neighbors in the conflict graph. Throughout this section, we
consider the conflict graph Gh at time t under channel state
h, and omit the subscripts t and h if there is no confusion.
3.1 Algorithm description
We assume that each time slot has two parts: contention
and transmission. The contention part has several intervals,
and each interval is further divided into mini-slots. We determine a feasible schedule during the contention part, and
with the computed schedule, transmits actual data during
the transmission part.
At a time slot, let B denote the feasible schedule (independent set of vertices) chosen by our algorithm. We explain
our solution, starting with an empty set and add vertices
to B by executing an iterative algorithm as shown in Algorithm 1.
At each interval i, some vertices are ‘determined’ as to
whether they belong to set B or not. Specifically, vertices in
Bi are ‘determined’ to be in B at interval i, and vertices in
(I(Bi )\Bi ) are ‘determined’ not to be in B at interval i. Let
Vi denote the set of vertices that have not been determined
yet at the beginning of interval i, i.e.,
Vi := V \ ∪i−1
j=1 I(Bj ) ,
which can be rewritten in a recursive form as
Vi = Vi−1 \I(Bi−1 ).
We say that a vertex in Vi is eligible at interval i. Let Ai ⊂ Vi
denote the set of vertices that will be ‘determined’ during
interval i, from which we will compute Bi . We will soon
see how to find Ai and Bi . From the definitions, it is clear
Algorithm 1 DistGreedy algorithm.
V0 ← V , B0 ← ∅
1: for i = 1 to logα β|V | do
2:
Vi ← Vi−1 \I(Bi−1 )
3:
Ai ← ∅
4:
for each v ∈ Vi do
5:
calculate w̄v (Vi ) := maxx∈Vi ∩I(v) wx
6:
if wv ≥ w̄vα(Vi ) then
7:
Ai ← Ai ∪ {v}
8:
end if
9:
end for
10:
Bi ← dist maximal matching(Ai )
11: end for
X
W*
L1
X
W*/a
L2
X
W*/a2
L3
*
3
W /a
Ĕ...
A1
that V0 = V and B0 = ∅. Finally, we have a couple of
configuration parameters α, β that will be explained later.
The algorithm can be described as follows. We start with
the entire set of vertices V0 = V and an empty set B0 = ∅.
Suppose that each vertex v knows its neighbors’ weights.
In wireless networks, this can be obtained by piggybacking/overhearing the information exchange or by explicitly
exchanging control messages.
1. At each interval i, the set of eligible vertices Vi is updated by excluding I(Bi−1 ) from Vi−1 (line 2 in Algorithm 1), where Bi−1 denotes the set of vertices that
are chosen during interval i−1 and I(Bi−1 ) denotes the
set of neighbors for Bi−1 (including Bi−1 itself). For
this purpose, each vertex that belongs to Bi−1 should
notify its neighbors by broadcasting a control message
during interval i − 1. (See Step 4 below.)
2. Each vertex v in Vi calculates its local maximum weight
w̄v (Vi ) := maxx∈Vi ∩I(v) wx from the weight information of its neighbors (in Vi ). Then each vertex v sets
itself as one of Ai if wv ≥ w̄v (Vi )/α, where α > 1.
(Lines 5 − 8.)
3. On the set Ai , we compute a maximal matching in a
distributed manner (line 10), which requires O(δ log2 |V |)
complexity [26] or O(δ) complexity with precomputation [14], where δ is the vertex degree. Let Bi denotes
the obtained set.
4. In the process of computing the maximal matching,
each neighbor of vertex v ∈ Bi should be informed
that v belongs to Bi . Hence, the vertices in I(Bi ) will
not participate in the next interval.
5. The above procedure repeats for J times, where J :=
⌈logα β|V |⌉. The set B(= ∪i Bi ) of vertices will be
returned as the final result. This is the set of links
that will transmit data packets during the time slot.
We have two configuration parameters α and β that will be
further discussed in the next section.
Note that our distributed greedy (DistGreedy) algorithm
requires O(δ) complexity at each interval and will run for
O(log |V |) intervals (using the algorithm in [14]). Hence,
the worst-case complexity will be O(δ log |V |). In some applications, e.g., regular topologies, δ is a fixed constant.
Thus, DistGreedy can be implemented with O(log |V |) complexity (or with polylogarithmic complexity in random net-
X
B1
X S*
V2=V1\I(B1)
Figure 1: Conflict graph with vertices and edges in
the layered format, where vertices are partitioned
into layers according to their weights.
works1 ), which is much lower than the O(|V |) complexity of
the known distributed implementation of GMS [4, 10].
Remarks: Note that the previous greedy approximations
to the MWIS problem shown in [24] have a similar iterative
procedure as DistGreedy. However, their algorithm works
vertex-by-vertex sequentially, which results in linear complexity in the worst case (e.g., consider a ring topology such
that, starting from a link, the link weights decrease in a
clockwise direction). Further, they have a different rule for
selecting a vertex at each interval, e.g.,P
they select vertex v
wv
wx
with the largest |Vi ∩I(v)|
or with wv ≥ x∈Vi ∩I(v) |Vi ∩I(x)|
.
This selection rule is the key to achieve the provable approximation ratio of 1δ . Unlike this previous work [24], DistGreedy reduces the complexity significantly by considering
multiple vertices in parallel and do not follow the strict sequential ordering. Further, the procedure stops after a certain number of intervals. At each interval, DistGreedy selects the vertices v with wv ≥ maxx∈Vi ∩I(v) wx /α. The end
1
result is a much better approximation ratio (≈ ∆
) and a
much better complexity (O(log |V |)). However, the parallel
processing also makes it more difficult to analyze the performance of DistGreedy. Nonetheless, in the next section,
we show that due to the selection rule of DistGreedy, it can
1
.
achieve the approximation ratio arbitrarily close to ∆
3.2 Performance Analysis
We evaluate the performance of our distributed greedy
1
(DistGreedy) algorithm, and show that it is in fact a ∆
approximation to the optimal solution. Motivated by [28],
we divide the vertices into layers L1 , L2 , . . . based on the
ratio of their weight to the maximum weight w∗ , as 2
∗
w∗
w
(3)
Li = v ∈ V i < wv ≤ i−1 .
α
α
1
In random networks, it is well-known that δ ∼ O(log |V |)
to ensure connectivity [9].
2
The algorithm in [28] computes a maximal matching for
each layer, and thus requires for each node to know which
layer it belongs to, or equivalently to know w∗ . However,
knowing the maximum weight w∗ may take O(|V |) time to
propagate in the worst case. In contrast, DistGreedy works
with local weight information and the layering structure is
only for the purpose of analysis.
Fig. 1 illustrates an example conflict graph in the layered
format.
We start our analysis with the following lemmas.
Lemma 1. For i ≤ logα β|V |, if vertex v ∈ Li , then
v ∈ I(∪ij=1 Bj ), and thus
Li ⊂ I(∪ij=1 Bj ).
(4)
Proof. If each vertex v ∈ Li selects itself for distributed
maximal matching no later than the i-th interval, i.e., if
v ∈ Li implies v ∈ ∪ij=1 Aj , then we can obtain the lemma,
since
v ∈ Li ⇒ v ∈ ∪ij=1 Aj ⇒ v ∈ ∪ij=1 I(Bj ) ⇒ v ∈ I(∪ij=1 Bj ),
(5)
where the second step comes from the fact Aj ⊂ I(Bj ), since
Bj is a maximal matching on Aj .
Now what remains to be shown is that v ∈ Li implies
v ∈ ∪ij=1 Aj . We show this by induction. It is clear that
when
i = 1, all vertices v ∈ L1 belong to A1 , because wv >
w∗
.
Suppose
that the statement is true for all i ≤ c. Note
α
that all vertices in ∪cj=1 Aj are not eligible at interval c + 1
since each vertex in Aj belongs to I(Bj ) for j = 1, 2, . . . , c
under our algorithm. Hence, at interval c + 1, no vertex in
∪cj=1 Aj is eligible, which immediately implies that no vertex
in ∪cj=1 Lj is eligible since Li ⊂ ∪ij=1 Aj for all i ≤ c. Now,
if there is a vertex v ∈ Lc+1 eligible at interval c + 1, i.e.,
v ∈ Vc+1 , then vertex v ∗should be included in Ac+1 , since all
vertices e with we > w
(i.e., e ∈ ∪cj=1 Lj ) are not eligible.
αc
Hence, the induction hypothesis must hold for i = c + 1.
This completes the proof.
Lemma 1 states that under DistGreedy, any vertex in layer
Li will be ‘determined’ after interval i ends. In the following lemma, we show that each vertex in layer i must have a
neighboring vertex that has a similar or higher weight and
that is chosen by DistGreedy after interval i ends. Combining these two lemmas, we can show that after interval i,
every vertex in Li or above is a neighbor of a vertex that is
already chosen by DistGreedy.
Lemma 2. For each vertex x ∈ Li (with i ≤ logα β|V |),
there exists v ∈ ∪ij=1 Bj such that x ∈ I(v) and α · wv ≥ wx .
Proof. From Lemma 1, we have x ∈ I(∪ij=1 Bj ), and
thus there exists v ∈ ∪ij=1 Bj such that x ∈ I(v). Let k ≤ i
be the smallest index such that x ∈ I(Bk ). Then, x ∈
/
I(∪k−1
j=1 Bj ) and there exists v ∈ Bk with x ∈ I(v). Since
v ∈ Bk ⊂ Ak , it should satisfy w̄v (Vk )/wv ≤ α from line
6 of Algorithm 1. Also since x ∈ I(v) and x is eligible at
interval k (because x ∈
/ I(∪k−1
j=1 Bj )), we have w̄v (Vk ) ≥ wx .
Hence, we obtain that α · wv ≥ wx .
Recall that S ∗ denotes the maximum weighted independent set over V . We define Di (v) as the set of vertices in
S ∗ ∩ Li that are connected to v by an edge in the conflict
graph, i.e.,
Di (v) := {x | x ∈ S ∗ ∩ Li , and x ∈ I(v)}.
(6)
Then |Di (v)| denotes the number of vertices selected by the
MWIS solution in layer Li that conflicts with v. The following lemma shows that the weight sum for S ∗ within layer
Li can be bounded by the weight sum for the independent
set chosen by DistGreedy up to interval i, multiplied by
α · |Di (v)|.
Lemma 3. At each interval i, we have
X
X
α · |Di (v)| · wv ≥
wx .
(7)
x∈Li ∩S ∗
v∈∪ij=1 Bj
Proof. From Lemma 2, we have that for each vertex
x ∈ S ∗ ∩ Li , there exists v ∈ ∪ij=1 Bj such that x ∈ I(v)
and α · wv ≥ wx . However, multiple x may map to the same
v. Nonetheless, for each of such v, at most |Di (v)| vertices
in S ∗ ∩ Li can potentially be neighbor of v in the conflict
graph. Therefore, we can obtain the result.
By summing both sides of (7) for all i, we can bound
the maximum weight sum by the weight sum of the vertices chosen by DistGreedy within a constant factor α∆.
(See the proof of Lemma 4 below.) However, if we were to
terminate after all vertices are considered, it would have resulted in O(|V |) complexity (e.g., consider a fully connected
1
1
graph with vertices whose weights are 1, α+ǫ
, (α+ǫ)
2 , ...). In
the next lemma, we show that even if DistGreedy stops after O(log |V |) intervals, the performance loss would still be
negligible.
Lemma 4. By setting α → 1 and β sufficiently large,
1
-approximation algorithm.
Algorithm 1 is a ∆(h)
Proof. Let B := ∪Jj=1 Bj , where J := ⌈logα β|V |⌉. By
summing (7) from i = 1 to J, we can obtain that
J
X
X
wx ≤
i=1 x∈Li ∩S ∗
J
X
X
α · |Di (v)| · wv
i=1 v∈∪i
j=1 Bj
≤
J
XX
α · |Di (v)| · wv
(8)
v∈B i=1
≤
X
α · ∆ · wv .
v∈B
Also, for i > J, we can obtain that
∞
X
X
wx ≤
i=J +1 x∈Li ∩S ∗
∞
X
X
wx ≤ |V | ·
i=J +1 x∈Li
w∗
w∗
≤
,
αJ
β
(9)
where w∗ denotes the largest weight among all the vertices.
The last inequality holds since J = ⌈log α β|V |⌉.
Combining (8) and (9), we can obtain that
α∆
X
wv +
v∈B
X
w∗
≥
wx .
β
x∈S ∗
(10)
Thus our result follows.
It has been shown in non-fading environments that a scheduling solution that is a γ-approximation to the MWIS problem
at each time slot can achieve at least γ fraction of the optimal
throughput [15,18]. It is straightforward to extend the result
to fading environment: a scheduling solution that is a γ(h)approximation to the MWIS problem under channel state h
at each time slot can achieve at least minh∈H γ(h) fraction
of the optimal throughput. Combining it with Lemma 4, we
can obtain the following Proposition.
Proposition 5. A scheduling solution that executes DistGreedy at each time slot can achieve ∆1∗ fraction of the optimal throughput, where ∆∗ = maxh∈H ∆(h).
0
1
2
3
10000
=1.1, 33 intervals
=1.5, 7 intervals
8
12
5
9
13
6
10
14
7
11
15
8000
Total queue lengths
4
=2.0, 4 intervals
=2.5, 3 intervals
=3.0, 2 intervals
6000
4000
2000
Figure 2: Grid network topology.
0
0.0
Remarks: In developing a distributed low-complexity GMS
algorithm, one of the main difficulties lies in the requirement
of the strict global ordering of selected vertices. For example, if the conflict graph is a linear graph, where the weights
of vertices monotonically decrease from the left to the right,
then the selection of the right-most vertex v with the smallest weight can be made only after the selection of its left-side
neighbor u, which again can be made only after the selection of the left-side neighbor of vertex u due to the linear
topology. This implies that the selection of the right-most
vertex v needs to be made after O(|V |) time.
Our result implies that the strict global ordering in the
GMS algorithm is not required for high performance. A
loose ordering would be sufficient, which can result in significant complexity reduction with negligible performance
degradation. We highlight that the state-of-the-art “dis1
tributed” ∆(h)
-approximation algorithm requires O(|V |) complexity [4, 10], while our local greedy algorithm significantly
lowers the complexity to O(δ(h) log |V |).
4.
NUMERICAL RESULTS
We evaluate DistGreedy, Greedy, and MaxWeight through
simulations, where MaxWeight is the optimal scheduler that
solves the MWIS problem at each time slot. We simulate
two networks: one with a grid topology and the other with
a randomly generated topology.
We first consider a grid topology as shown in Fig. 2. Each
link has an average transmission rate of one, two, or three
packets per time slot, which are signified in the figures by
the number of lines between two nodes, e.g., one line implies
one packet per time on average. At each time slot, actual
link rate changes and is chosen uniformly at random from
the range [0, 2(Avg. rate)]. Since DistGreedy approximates
the optimal solution to the MWIS problem at each time slot,
we focus on the behavior of DistGreedy under static interference models, where the interference relationship does not
change across time. In particular, we use one-hop (or primary) interference model, under which two links that share
a node cannot transmit at the same time. We impose singlehop traffic of load ρ on every link: at each time slot, each
link has a packet arrival with probability ρ. The arrivals
are i.i.d. across time slots and links. We set DistGreedy to
have ⌈logα β|V |⌉ intervals at each time slot. We use a linkcoloring technique to find a maximal matching, under which
(δ + 1) mini-slots are sufficient [14]. Since δ = 6 in our grid
topology, we use 10 mini-slots at each interval. The number
of mini-slots are not taken into account in the performance
measurements. Each result is an average of 10 simulation
runs for 106 time slots.
0.2
0.4
0.6
0.8
Figure 3: Performance of DistGreedy with different
α.
Fig. 3 illustrates the performance of DistGreedy in terms
of total queue lengths with different α settings and β = 1.
Sharp increases of queue lengths imply the boundary of the
capacity region. Note that a larger α means thicker layers
and thus a smaller number of intervals. The results show
that the performance is not sensitive to the value of α in a
range [1.1, 3] and a small number of intervals (e.g., α = 2.0)
would be sufficient for high throughput performance.
While running DistGreedy, we also trace the maximum
weight sum, the weight sum of the schedule selected by DistGreedy, and the weight sum of the schedule that would be
chosen by Greedy, at each time slot. Fig. 4 depicts the ratio
of each weight sum (from DistGreedy and Greedy) to the
maximum weight sum. It shows that both GMS and DistGreedy typically achieve much higher ratios than the analytical bound, which is 12 in the one-hop interference model
and shown in the figure using a dotted line. Also, as α gets
closer to 1, DistGreedy algorithm approaches Greedy algorithm because layers become narrower and the number of
intervals increases.
In Fig. 5, we compare the performances of MaxWeight,
Greedy, DistGreedy (with α = 2, β = 1), and Q-CSMA,
where Q-CSMA is a CSMA algorithm known to be throughputoptimal in non-fading environments. For Q-CSMA, we use
the version shown in [8], i.e., each link v sets its access prob1
ability for the decision vector to |I(v)|
and sets its weight for
h
link activity to log(Ql (t) · rl (t)). The results in Fig. 5 illustrate that Q-CSMA, which has the lowest complexity O(1)
among the scheduling schemes, has much poor throughput
and delay performance than the others. In particular, its delay grows quickly at an offered load much lower than other
algorithms, which suggests that it is not throughput optimal in fading environments. In contrast, DistGreedy has
similar queue lengths to MaxWeight and Greedy. In other
words, it empirically achieves similar throughput and delay
performance to the optimal. Further, the simulation results
suggest that the actual performance of DistGreedy could
significantly outperforms the analytical lower bounds.
Next we consider a network that is randomly generated.
We place a total of 32 nodes at random within 1×1 area. We
connect two nodes with a link if they are within a distance
of 0.25. Each link has a time-varying link rate, which is
randomly chosen in the range of [0, 10] packets per slot, and
i.i.d. across links and time. We generated single-hop traffic
over 24 links, which are chosen at random. Each source
Ratio of weight sum to the maximum
1.0
502
Client 1
Server
A
510
0.4
Greedy
0.0
0
200
400
600
800
1000
Ratio of weight sum to the maximum
1.0
0.8
0.6
0.4
Greedy
DistGreedy
0.2
0.0
0
200
400
600
800
1000
time
(b) α = 1.1
Figure 4: Ratio of the achieved weight sum to the
maximum weighted sum.
10000
MaxW eight
Greedy
DistGreedy
Total queue lengths
Q-CSMA
6000
4000
2000
0
Figure 5:
schemes.
0.2
0.4
0.6
0.8
Performance comparison of scheduling
injects a random number of packets in the range of [1, 5] at
each time slot, with probability ρ. The results are similar
to the grid network and are omitted due to lack of space.
We can observe that the throughput and delay performance
of DistGreedy are close to those of MaxWeight and Greedy,
even though it has a significantly lower complexity.
5.
B
509
505
507
Client 2
DistGreedy
0.2
(a) α = 2.0
0.0
504
Client 3
0.6
time
8000
503
0.8
PRELIMINARY EXPERIMENT RESULTS
In this section, we provide preliminary experimental results and show that the opportunistic gains of wireless fading can be achieved in practice. We have implemented a
version of DistGreedy in hardware driver by modifying the
medium access control of IEEE 802.11 DCF. Specifically,
we use the ath5k device driver [2] over Voyage Linux [3] installed into the Alix 2D2 system board [1]. We modified it
such that each node maintains information of recent queue
lengths and transmission rates (link capacity) of neighboring nodes, and that a header in the data frame includes the
queue length information of the transmitter. The transmis-
Figure 6: Experiment setup.
sion rate information can also be obtained from the device if
the frame is successfully received. Each node who receives or
overhears the frame updates the queue and rate information
of the transmitter from the header.
Unlike IEEE 802.11 DCF, DistGreedy in Algorithm 1 requires time synchronization between nodes. We avoid the
synchronization overheads by implementing an approximation of our algorithm while capturing the essential feature
of the algorithm. Further, while the DistGreedy scheme
is a link-based algorithm, current implementation of IEEE
802.11 DCF operates on nodes. Due to this difference, we
use wn′ (t) and w̄n′ (t) to denote node n’s weight and its local
maximum weight at time t, respectively. Our DistGreedy
implementation works just like IEEE 802.11 DCF, except
that when the transmitter n has a frame to send (asynchronously with other nodes) at time t, it calculates the local maximum weight w̄n′ (t) from the information that it has
w̄′ (t)
received from its neighbors, and estimates θ := ⌊ α·wn′ (t) ⌋.
n
Then, it chooses the contention window size at random within
[0, 7] if θ = 0, within [0, 31] if θ = 1, within [0, 63] if θ = 2,
and within [0, 127] otherwise3 . In this way, our implementation effectively has four intervals. Moreover, our implementation does not compute a maximal matching, which
reduces the complexity further (compared with Algorithm 1,
line 10). We set α = 10/9, the maximum buffer size to 100
frames, and the IP packet size to 512 bytes. We remark that
finding an optimal number of intervals and the contention
window sizes is an interesting open question but beyond the
scope of the paper.
Fig. 6 shows our experiment setting at the 5th floor of the
ECE department building in UNIST, South Korea. We set
three stationary clients and let each client transmit data at 3
Mbps to the single mobile server. All the clients can overhear
each other’s transmissions. The server moves between two
positions A and B as shown in Fig. 6, at every 60 seconds,
starting from A. Clearly, Client 1 has a good channel state
when the server locates in Position A, and Client 3 performs
well when the server locates in Position B. We conduct our
experiments for 5 minutes and measure transmission rate,
queue lengths, and throughput of each client.
Figs. 7 and 8 illustrate the experiment results. Due to high
measurement variations, we show time-averaged values using
exponential weighted moving average. Figs. 7(a) and 8(a)
show that the variations of the total (average) capacity of
three clients is less than the variations of the transmission
rate of an individual link, which is significant especially when
the server moves. Note that transmission rate of a link has
a discrete value of {6, 9, 12, 24, 36, 48, 56} Mbps. Instantaneous link rate changes very frequently across time though
3
Non-overlapped windows for each θ seems to be a more
intuitive choice. Unfortunately, we are unable to select a
window that starts with non-zero value due to configuration
restriction in our device firmware.
200
120
90
DistGreedy
802.11
60
30
0
10
802.11
DistGreedy
Total throughput (Mbps)
Total queue length (pkts)
Total data rate (Mbps)
150
150
100
50
8
6
4
2
0
0
60
120
180
240
300
DistGreedy
802.11
0
0
60
Time (sec)
120
180
240
300
0
60
Time (sec)
(a) Total transmission rate
120
180
240
300
Time (sec)
(b) Total queue length
(c) Total throughput
Figure 7: Overall performance across all clients. Total transmission rate (a) shows that the total (average)
link rate sum is similar. However, total queue lengths (b) and throughputs (c) clearly show that DistGreedy
outperforms the IEEE 802.11 DCF.
36
24
18
12
9
6
DistGreedy
802.11
0
60
120
180
240
Time (sec)
(a) Transmission rate
4
DistGreedy
802.11
Throughput (Mbps)
Queue length (pkts)
Data rate (Mbps)
100
54
48
75
50
25
0
300
DistGreedy
802.11
3
2
1
0
0
60
120
180
240
300
0
Time (sec)
(b) Queue length
60
120
180
240
300
Time (sec)
(c) Throughput
Figure 8: Performance of Client 3. Similar results as in Fig. 7 are observed when we focus on the performance
of a client.
these changes are not shown in the figures due to averaging.
Further the link rate is chosen by the hardware physical layer
depending on the channel state at a given time. We have
observed that transmission rate changes frequently and often in an unpredictable manner. Since the transmission rate
is not under our control, it is difficult to maintain exactly
the same channel environment when we compare two different MAC protocols. This is the reason that DistGreedy
and IEEE 802.11 DCF have different link rates in Figs. 7(a)
and 8(a). Nonetheless, the figures show that overall link
rates are similar for both cases. We may force the hardware
to use a fixed transmission rate at the transmitter. However,
in such a case, we may see frequent transmission failures due
to insufficient SINR at the receiver.
Fig. 7 shows that our DistGreedy implementation achieves
better network performance in terms of reduced total queue
length (Fig. 7(b)) and total throughput performance (Fig. 7(c))
under similar channel states. Indeed, DistGreedy implementation maintains queue lengths low for all the three clients,
and IEEE 802.11 DCF results in many drops due to buffer
overflows. For example, the queue length of Client 3 frequently increases up to 100 under IEEE 802.11 DCF even
through it does not show up in Fig. 8 due to exponential
weighted time averaging.
In Figs. 8(a) and 8(b), we can observe that under IEEE
802.11 DCF, high link rate (i.e., in [60, 120] and [180, 240])
does not lead to low queue lengths, since IEEE 802.11 DCF
does not opportunistically exploit wireless fading. In contrast, it shows that the DistGreedy implementation successfully keeps the queue length low, especially when the link
rate is high. This implies that DistGreedy can account
for the channel variations, and thus takes the advantage
of the opportunistic gains. Another interesting observation
is that in Fig. 8(c), IEEE 802.11 DCF often suffers from
poor throughput performance when the server moves, i.e.,
after 120 and 240 seconds, while DistGreedy implementation
maintains high throughput performance during the moves.
6. CONCLUSION
In this paper, we develop a distributed scheduling scheme
that is provably efficient under wireless fading. By taking a
local greedy approach, we prove that our scheme is a ∆1∗ approximation to the Maximum Weighted Independent Set
problem, where ∆∗ is the maximum conflict degree, and
has O(log |V |) complexity (or polylogarithmic complexity),
where |V | is the number of links.
We evaluate our scheme through simulations in grid networks and random networks. The results show that our
distributed scheduling scheme is insensitive to parameter
settings, and achieves throughput and delay performance
similar to those of the optimal solution. We also implement
our scheme with hardware by modifying the existing IEEE
802.11 DCF. The experimental results show that our modification results in better throughput performance with low
queue length by taking into account time-varying link capacities.
7. ACKNOWLEDGMENTS
This work has been partially supported by NSF grants
CNS-1012700 and CNS-0643145, and a MURI grant from
the Army Research Office W911NF-08-1-0238, and in part
by the Basic Science Research Program through the NRF of
Korea, funded by the Ministry of Science, ICT, and Future
Planning (No. 2011-0008549).
8.
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